Relation between Roots and Tangent Lines of Function in Fractional Dimensions: A Method for Optimization Problems
Authors: Ali Dorostkar
In this paper, a basic schematic of fractional dimensional optimization problem is presented. As will be shown, a method is performed based on a relation between roots and tangent lines of function in fractional dimensions for an arbitrary initial point. It is shown that for each polynomial function with order N at least N tangent lines must be existed in fractional dimensions of 0 < α < N+1 which pass exactly through the all roots of the proposed function. Geometrical analysis of tangent lines in fractional dimensions is also presented to clarify more intuitively the proposed method. Results show that with an appropriate selection of fractional dimensions, we can directly find the roots. Method is presented for giving a different direction of optimization problems by the use of fractional dimensions.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.3455723Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 269
 J. Noceda, S.J. Wright, Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2006.
 I. Newton, Methodus ﬂuxionum et serierum inﬁnitarum, 1664-1671.
 P. Sebah, X.Gourdon, Newton’s method and high order iterations, 2001. Available from: computation.free.fr/Constants/constants.html.
 G. Adomian, R. Rach, "On the solution of algebraic equations by the decomposition method", Math. Anal. Appl. Vol. 105, pp.141-166, 1985.
 G. Adomian, Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, Dordrecht, 1994.
 K. Abbaoui, Y. Cherruault, "Convergence of Adomian method applied to nonlinear equations," Math. Comput. Model. vol.20, no.9, pp. 69-73, 1994.
 E. Babolian, J. Biazar, "Solution of nonlinear equations by modiﬁed Adomian decomposition method," Appl. Math. Comput. vol.132, pp. 167-172, 2002.
 K. Oldham and J. Spanier, Fractional Calculus, Academic Press, New York, 1974.
 K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Differential Equations, John Wiley & Sons, New York,1993.
 I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
 P.J. Torvik, R.L. Bagley, "On the Appearance of the Fractional Derivative in the Behavior of Real Materials," ASME. J. Appl. Mech., vol.51, no.2, pp. 294-298, 1984.
 R. Almeida, A. B. Malinowska and D. F. M. Torres, "A fractional calculus of variations for multiple integrals with application to vibrating string," J. Math. Phys. Vol.51, no. 3, 033503, 12 pp.arXiv:1001.2722, 2010.
 N. R. O. Bastos, R. A. C. Ferreira and D. F. M. Torres, "Necessary optimality conditions for fractional difference problems of the calculus of variations," Discrete Contin. Dyn. Syst. Vol. 29, no. 2, pp.417–437. arXiv:1007.0594, 2011.
 T. Odzijewicz, A. B. Malinowska and D. F. M. Torres, "Fractional variational calculus of variable order, Advances in Harmonic Analysis and Operator Theory," The Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck), Operator Theory: Advances and Applications, vol. 229, pp. 291–301. arXiv:1110.4141, 2013
 O.P. Agrawal, "Fractional variational calculus and the transversality conditions," J. Phys. A, 39:10375-10384, 2006.
 U. N. Katugampola, "New approach to a generalized fractional integral," Appl. Math. Comput. 218 (2011), no. 3, 860–865.
 R. Almeida, A. B. Malinowska and D. F. M. Torres, "Fractional Euler-Lagrange differential equations via Caputo derivatives, In: Fractional Dynamics and Control," Springer New York, part 2, pp. 109–118. arXiv:1109.0658,2012
 A. A. Kilbas and M. Saigo, "Generalized Mittag-Leffer function and generalized fractional calculus operators, Integral Transform," Spec. Func. vol. 15, no. 1, pp.31–49, 2004.
 A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, "Theory and applications of fractional differential equations," Elsevier, Amsterdam, 2006.
 M. Klimek, "On solutions of linear fractional differential equations of a variational type," The Publishing Office of Czestochowa University of Technology, Czestochowa, 2009.
 N. H. Sweilam, M. M. Khader and R. F. Al-Bar, "Numerical studies for a multi-order fractional differential equation," Physics Letters A, vol. 371, pp. 26-33, 2007.